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Unformatted text preview: Mathematical Methods I Natural Sciences Tripos, Part IB Dr Gordon Ogilvie Michaelmas Term 2008 Example Sheet 1 1. (a) To what quantities do the following expressions in suffix notation (using the summation convention) correspond? Simplify where appropriate. ii , ij a j , ij a i a j , ij ij , iji , ijk jk , b i ijk a k c j , ijk A 3 i A 1 k A 2 j . (b) For each of the following equations, either give the equivalent in vector or matrix notation, or explain why the equation is invalid. x i = a i b k c k + d i , x i = a j b i + c k d i e k f j , u = ( jkl v k w l ) x j , ijk x j y k ilm x l y m = 1 , A ik B kl = T ik kl , x = A i B i y . (c) Write the following equations in suffix notation using the summation conven- tion. ( x + y ) ( x y ) = 0 , x = | a | 2 b | b | 2 a , (2 x y ) ( a + b ) = . 2. (a) Show that a ( b c ) = b ( c a ) = c ( a b ) . (b) Explain why ijk klm = il jm im jl . Hence find ijk ijk . 3. A fluid flow has the constant velocity vector (in Cartesian coordinates) v ( r ) = (0 , , W ) . Explicitly calculate the volume flux of fluid, Q = integraldisplay v d S , flowing across (a) the open hemispherical surface r = a , z greaterorequalslant 0, and (b) the disc r lessorequalslant a , z = 0. Verify that the divergence theorem holds. 4. For a surface S enclosing a volume V , apply the divergence theorem to a vector field F = a p , where a is an arbitrary constant vector and p ( r ) is a scalar field. Deduce that integraldisplay V ( p ) d V = integraldisplay S p d S ....
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This note was uploaded on 05/09/2009 for the course DAMTP NST 1B Mat taught by Professor Gogilvie during the Spring '09 term at Cambridge.
- Spring '09